We introduce new metric structures on a smooth manifold (called “weak" structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures (φ,ξ,η,g) and allow us to take a fresh look at the classical theory and find new applications. This assertion is illustrated by generalizing several well-known results. It is proved that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. It is shown that a weak almost contact structure with parallel tensor φ is a weak cosymplectic structure and an example of such a structure on the product of manifolds is given. Conditions are found under which a vector field is a weak contact vector field.

In this paper, we introduce the notions of nearly Sasakian and nearly Kähler statistical structures with a non-trivial example. The conditions for a real hypersurface in a nearly Kähler statistical manifold to admit a nearly Sasakian statistical structure are given. We also study Invariant and anti-invariant statistical submanifolds of nearly Sasakian statistical manifolds. At the end, some conditions under which such a submanifold of a nearly Sasakian statistical manifold is itself a nearly Sasakian statistical manifold are given.

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly K\"{a}hlerian structures) and study their geometry. We introduce weak nearly K\"{a}hlerian manifolds (generalizing nearly K\"{a}hlerian manifolds) and characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds.

A class of nearly Sasakian manifolds is considered. We discussion geometric effects of some symmetries on such manifolds, and show, under a certain condition, that the class of Ricci-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a nearly Sasakian space form with Ricci tensor satisfying the Codazzi equation is either a Sasakian manifold with a constant $\phi$-holomorphic sectional curvature $\mathcal{H}=1$ or a $5$-dimensional proper nearly Sasakian manifold with a constant $\phi$-holomorphic sectional curvature $\mathcal{H}>1$. We also prove that the spectrum of the operator $H^{2}$ generated by the nearly Sasakian manifold is a set of simple eigenvalue $0$ and an eigenvalue of multiplicity $4$. We show that there exist integrable distributions on the same manifolds with totally geodesic leaves, and prove that there are no proper nearly Sasakian space forms with constant sectional curvature.

This is the fourth part of [6] on the existence of K¨ahler Einstein metrics of the general type I almost homogeneous manifolds of cohomogeneity one. We actually carry out all the results in [8] to the type I cases. In part II [14], we obtained a lot of new K¨ahler-Einstein manifolds as well as Fano manifolds without K¨ahler-Einstein metrics. In particular, by applying Theorem 15 therein, we have complete results in the Theorems 3 and 4 in that paper. However, we only have some partial results in Theorem 5 there. In this note, we shall give a report of recent progress on the Fano manifolds Nn,m when n > 15 and N′n,m when n > 4. We actually give two nice pictures for these two classes of manifolds. See our Theorems 1 and 2 in the last section. Moreover, we post two conjectures. Once we could solve these two conjectures, the question for these two classes of manifolds would be completely solved. With applying our results to the canonical circle bundles we also obtain Sasakian manifolds with or without Sasakian-Einstein metrics. That also give some open Calabi-Yau manifolds.